It’s clear that the last series contains precisely the same terms as the preceding one. The regrouping and the rearranging of the terms is allowable, since also (1) is converges absolutely. In fact, if one would multiply the series of exsuperscriptexe^{x} with the series y+y33!+y55!+…ysuperscripty33superscripty55normal-…y\!+\!\frac{y^{3}}{3!}\!+\!\frac{y^{5}}{5!}\!+\!... of sinh⁡yy\sinh{y} (which converges absolutely ∀x∈ℝfor-allxℝ\forall x\in\mathbb{R}), one would get the series like (1) but all signs “+”; by the Cauchy multiplication rule this series converges especially for each positivexxx and yyy, in which case it is a series with positive terms; hence (1) is absolutely convergent.